On the accuracy of implicit neural representations for cardiovascular anatomies and hemodynamic fields
SIREN, MFN-Gabor, and MHE compressed thoracic-aorta fields up to 230×, with ~1 mmHg pressure and sub-1.6 mm anatomical errors.
Key Findings
Methodology
The study benchmarks nine INR designs that map coordinates to pressure, velocity, or signed distance. Compared systems include Tanh-MLP, fixed and trainable positional encoding, SIREN, MFN-Fourier, MFN-Gabor, and multiresolution hash encoding (MHE). Training is largely standardized: five layers, 512 neurons per layer, 10,000 iterations, batch size 1024, and learning rate 1e-4. For mesh-based field queries, neural fitting at unstructured tetrahedral nodes is combined with local shape-function interpolation.
Key Results
- For realistic space-time thoracic-aorta fields, INRs reached compression ratios of approximately 230, while maximum absolute pressure errors were about 1 mmHg and velocity errors were 5–10 cm/s, without extensive hyperparameter tuning.
- Across 48 thoracic-aortic anatomies, reconstructed signed-distance geometries had average absolute discrepancies below 0.5 mm and maximum discrepancies below 1.6 mm, indicating strong continuous geometric fidelity.
- SIREN, MFN-Gabor, and MHE were the strongest overall architectures. SIREN favors differentiable high-frequency representation, MFN-Gabor captures frequency plus locality, and MHE provides efficient multiscale encoding with a small downstream network.
Significance
The paper moves INR evaluation beyond images and graphics into physically meaningful cardiovascular simulation data. By quantifying pressure, velocity, geometry, compression, and interpolation behavior, it addresses a practical bottleneck in storing and reusing large finite-element results. Successful representations could accelerate visualization, optimization, uncertainty quantification, and digital-twin pipelines. Equally important, the study shows that compression quality must be judged jointly by numerical error, spatial locality, and physical use—not by file size alone.
Technical Contribution
The main technical contribution is a unified empirical benchmark spanning 2D–4D fields, idealized pipes, patient-specific aortas, and SDF anatomies. It contrasts several mechanisms for overcoming spectral bias: periodic activations, positional encodings, multiplicative Fourier/Gabor filters, and trainable hash grids. The node-fitting plus shape-function strategy is particularly pragmatic: the INR memorizes finite-element values, while interpolation supplies local evaluation and visualization. This creates an engineering bridge between continuous neural fields and established unstructured-mesh workflows.
Novelty
Unlike most related work centered on images, NeRF rendering, or static geometry, this study systematically evaluates INRs on realistic cardiovascular pressure and velocity fields while also testing anatomical SDFs. Its novelty is primarily evaluative and methodological rather than a new single architecture: it establishes a cross-dimensional, cross-physics comparison under consistent training settings and reports clinically interpretable errors, compression ratios, and geometry discrepancies.
Limitations
- The targets are numerical SimVascular finite-element solutions rather than direct clinical measurements. Consequently, the reported errors measure approximation to a simulator, not clinical predictive validity or robustness to imaging noise.
- Most architectures use standardized hyperparameters rather than exhaustive tuning. The best compression–accuracy trade-off may therefore change with field bandwidth, network capacity, resolution, or patient anatomy.
- High-capacity encodings promote memorization but can harm interpolation in small-data regimes; SDFs may also miss thin structures or difficult topologies.
Future Work
Future studies should incorporate 4D-MRI flow data, conservation and PDE constraints, and cross-patient validation. Promising directions include adaptive frequency selection, automated rate–distortion control, topology-preserving SDF losses, uncertainty quantification, and rapidly updated INRs for patient-specific digital twins. Comparisons should also include clinical measurement error and solver discretization error.
AI Executive Summary
Modern cardiovascular solvers can resolve coupled fluid, structural, and temporal phenomena, but their outputs are enormous. A conventional mesh stores values at many fixed locations, creating major burdens for visualization, optimization, uncertainty quantification, and digital twins. Implicit neural representations (INRs) offer an alternative: network weights encode a continuous function that can be queried at arbitrary coordinates. Yet their reliability for scientific hemodynamics has remained poorly characterized.
Lee and Schiavazzi benchmarked Tanh-MLPs, fixed and trainable positional encodings, SIREN, MFN-Fourier, MFN-Gabor, and multiresolution hash encoding (MHE). The target data came from SimVascular finite-element simulations of cylindrical pipes and patient-specific thoracic aortas, including steady and pulsatile pressure and velocity fields. Anatomies were represented separately with signed distance functions. To reduce the gap between memorization and useful interpolation, the authors fitted values at unstructured tetrahedral nodes and used local shape functions for extraction and visualization.
The results are striking but qualified. Thoracic-aorta space-time fields were compressed by up to approximately 230 times, with maximum pressure errors around 1 mmHg and velocity errors of 5–10 cm/s. Across 48 aortic anatomies, average and maximum geometric discrepancies remained below 0.5 and 1.6 mm. SIREN, MFN-Gabor, and MHE performed best overall, suggesting complementary benefits from periodic activations, localized spectral filters, and multiscale trainable encoding. The work supports INR-based scientific data compression, but clinical validation, physical conservation, cross-patient generalization, and preservation of thin anatomical structures remain essential before deployment.
Deep Analysis
Background
Cardiovascular computing has progressed from conceptual models to three-dimensional multiphysics simulations with hundreds of millions of degrees of freedom. SimVascular and finite-element solvers generate detailed pressure, velocity, and time-dependent anatomical fields, but storage and transfer increasingly constrain downstream analysis. INRs replace explicit voxels or meshes with continuous coordinate-to-field functions. Previous successes such as SIREN and NeRF largely emphasized images, rendering, or geometry; their accuracy for real hemodynamic fields was not established.
Core Problem
The precise question is whether modern INRs can represent high-frequency, localized, pulsatile cardiovascular fields accurately enough for scientific reuse. Standard MLPs exhibit spectral bias, learning smooth low-frequency content before boundary layers and branch-level details. The study also asks whether high memorization capacity damages interpolation away from training nodes, and whether the same representation paradigm can encode patient-specific anatomy through signed distance functions.
Innovation
- �� A unified evaluation from grayscale images and ideal pipes to patient-specific aortas.
- �� Direct comparison of periodic activations, fixed/trainable positional encoding, MFN-Fourier, MFN-Gabor, and MHE.
- �� Joint assessment of pressure, velocity, space-time fields, and SDF geometries.
- �� Integration of INR node fitting with tetrahedral shape-function interpolation.
- �� Reporting of compression ratios, maximum physical errors, and anatomical discrepancies rather than visual quality alone.
Methodology
- �� Inputs: normalized spatial coordinates XYZ, optionally time T; outputs: pressure, velocity components, or SDF.
- �� MLP/PE: Tanh networks receive Gaussian encoding γ1(x)=√2[cos(f1x),…,cos(fLx)] or powers-of-two sine/cosine features.
- �� SIREN: every layer uses sin(Wixi+bi), with first-layer ω0=30, supporting high frequencies and stable derivatives.
- �� MFN: Hadamard products recursively combine filters; Fourier uses sin(ωx+φ), whereas Gabor adds a localized Gaussian envelope.
- �� MHE: hierarchical hash grids use baseline L=16, D=2, log(T)=19, and Rmax=32, followed by SiLU layers.
- �� Training: five layers of width 512, 10,000 iterations, batch 1024, learning rate 1e-4. SDF zero level sets recover surfaces.
Experiments
Datasets include a cylindrical pipe of radius 2 cm and length 30 cm, with 2,354 nodes, 11,208 tetrahedra, and 160 reported time steps; a 256×186 zebra image; and Vascular Model Repository aorta model 0011H_AOH. Steady and pulsatile fields were tested in 1D–4D settings. Geometry experiments covered a sphere, pulmonary tree, thoracic aorta, and 48 additional patient-specific thoracic aortas. Metrics were compression ratio, maximum absolute pressure/velocity error, and mean/maximum SDF-derived anatomical discrepancy.
Results
The best thoracic-aorta field compression was approximately 230×, with pressure error near 1 mmHg and velocity error of 5–10 cm/s. For 48 anatomies, mean discrepancy was below 0.5 mm and maximum discrepancy below 1.6 mm. SIREN, MFN-Gabor, and MHE were consistently strongest overall. The comparison indicates that high-frequency periodic activations, localized Gabor kernels, and multiresolution hash features each mitigate spectral bias, although greater expressiveness can increase memorization and reduce out-of-sample interpolation.
Applications
INRs can serve as compact archives for finite-element results, enabling coordinate-based visualization, remote transmission, parameter optimization, and uncertainty studies. SDFs can support continuous vessel reconstruction, normal estimation, collision detection, surface extraction, and mesh generation. Practical deployment requires coordinate normalization, error certificates, physical residual checks, and interoperability with imaging and solver pipelines.
Limitations & Outlook
The benchmark approximates simulations rather than measured clinical flows, so it does not establish robustness to MRI noise, segmentation uncertainty, or model-form error. Uniform hyperparameters may understate architecture-specific optima, while the computational cost of training and encoding is not the same as inference cost. Memorization-oriented methods are problematic for small datasets, and SDFs can lose thin or topologically complex structures. Future work should combine clinical data, PDE constraints, topology-aware losses, adaptive compression, and cross-patient validation.
Plain Language Accessible to non-experts
Imagine a huge factory that records how water moves through a branching pipe. The old system writes down the pressure and motion at every location and every moment, like printing a separate instruction card for every screw in the building. The archive becomes enormous, and asking for a value at a new location is awkward.
An INR is like training one highly capable factory guide. You give the guide a location and a time, and it estimates what is happening there from the patterns stored inside its internal settings. Some guides mainly remember broad trends and miss tiny details near pipe branches. SIREN helps them notice rapid changes; MFN-Gabor records both the kind of pattern and where it occurs; MHE uses several map scales to locate information efficiently.
In the experiments, this guide compressed thoracic-aorta flow data by up to about 230 times. Pressure differed by roughly 1 mmHg at the worst point, while velocity differed by 5–10 cm/s. For 48 vessel shapes, the average mismatch was under 0.5 mm. That could make simulation archives easier to move and explore.
But the guide learned mainly from computer-generated factory records, not direct hospital measurements. It may struggle with noisy scans, unusual disease, tiny branches, or questions far from the examples used in training. Before clinical use, it must be checked against real data and the rules governing fluid motion.
ELI14 Explained like you're 14
Think about a video game with a living blood-vessel city. The slow method saves every map square, every instant, and every detail of how blood moves. The file gets gigantic. An INR trains a small AI to learn the city’s pattern instead. Give it coordinates and a time, and it predicts the pressure or direction there—almost like asking a game character what is happening at any point!
Here is the tricky part: ordinary neural networks often learn the big, smooth picture first, like finishing easy homework before attempting the hardest questions. Tiny changes near vessel branches may be ignored. SIREN uses wave-like sine functions to catch fast details. MFN-Gabor combines “what kind of pattern?” with “where is it?” MHE acts like a stack of maps with different zoom levels.
The researchers tested pipes, a zebra picture, a realistic thoracic aorta, and 48 more aortic shapes. The flow data became up to about 230 times smaller. The worst pressure error was around 1 mmHg, velocity error was 5–10 cm/s, and the average shape error was below 0.5 mm. Pretty impressive!
Still, this is not an all-knowing medical robot. The AI mostly learned simulated data, so real scans, noise, unusual diseases, and very thin vessels may surprise it. Future versions could learn from MRI and obey fluid rules. Then doctors might explore blood flow quickly, compare treatment plans, and build patient-specific digital twins.
Glossary
Implicit Neural Representation (INR)
An INR stores a continuous function in neural-network parameters instead of explicitly storing every voxel or mesh value. A coordinate query produces the field value at that location.
The paper uses INRs for pressure, velocity, and anatomical geometry.
Spectral Bias
Neural networks tend to learn low-frequency, smooth components before high-frequency details. This can obscure boundary layers, branch features, and sharp anatomical variation.
The benchmark tests activations and encodings designed to reduce this bias.
SIREN
A sinusoidal representation network whose layers use sine activations. It is infinitely differentiable and well suited to high-frequency fields and derivative-based equations.
The implementation uses first-layer frequency ω0=30.
MFN-Gabor
A multiplicative filter network recursively combines learned filters using elementwise products. Gabor filters provide both frequency selectivity and spatial localization.
It is one of the paper’s best-performing architectures for localized cardiovascular structure.
Multiresolution Hash Encoding (MHE)
MHE stores trainable features on hierarchical grids and interpolates nearby hashed vertices. Multiple resolutions improve multiscale expressiveness while controlling memory.
The baseline uses 16 levels, feature dimension 2, log(T)=19, and Rmax=32.
Signed Distance Function (SDF)
An SDF assigns signed distance to a surface: zero on the surface, negative inside, and positive outside. The zero level set therefore defines the geometry.
The paper reconstructs spheres, pulmonary trees, and aortas with SDFs.
Open Questions Unanswered questions from this research
- 1 Clinical reliability remains unresolved because simulated fields differ from noisy 4D-MRI measurements. Independent validation must separate representation error from imaging, segmentation, and model-form error.
- 2 No universal rate–distortion rule links network size, compression, physical error, and patient variability. A shared benchmark across diseases, scanners, and flow regimes is needed.
- 3 Thin vessels, complex topology, and queries far outside training samples may fail. Topology-aware losses and PDE constraints could address these cases.
Applications
Immediate Applications
Hemodynamic archive compression
Simulation groups can encode pressure and velocity fields as INRs, reducing storage and enabling coordinate-based visualization. Deployment should retain normalization metadata, weights, training-node checks, and explicit error reports against the original finite-element solution.
Continuous vessel reconstruction
Imaging pipelines can train SDFs, extract zero level sets, and compute smooth surface normals for meshing or shape comparison. Thin branches should be independently inspected because SDF representations may erase small structures.
Long-term Vision
Patient-specific cardiovascular digital twins
Future systems could combine imaging, measured flow, and PDE constraints in an INR that updates rapidly as new observations arrive. Clinical translation will require uncertainty estimates, regulatory validation, interoperability, and prospective outcome studies.
Abstract
Implicit neural representations (INRs, also known as neural fields) have recently emerged as a powerful framework for knowledge representation, synthesis, and compression. By encoding fields as continuous functions within the weights and biases of deep neural networks-rather than relying on voxel- or mesh-based structured or unstructured representations-INRs offer both resolution independence and high memory efficiency. However, their accuracy in domain-specific applications remains insufficiently understood. In this work, we assess the performance of state-of-the-art INRs for compressing hemodynamic fields derived from numerical simulations and for representing cardiovascular anatomies via signed distance functions. We investigate several strategies to mitigate spectral bias, including specialized activation functions, both fixed and trainable positional encoding, and linear combinations of nonlinear kernels. On realistic, space- and time-varying hemodynamic fields in the thoracic aorta, INRs achieved remarkable compression ratios of up to approximately 230, with maximum absolute errors of 1 mmHg for pressure and 5-10 cm/s for velocity, without extensive hyperparameter tuning. Across 48 thoracic aortic anatomies, the average and maximum absolute anatomical discrepancies were below 0.5 mm and 1.6 mm, respectively. Overall, the SIREN, MFN-Gabor, and MHE architectures demonstrated the best performance. Source code and data is available at https://github.com/desResLab/nrf.